Doob’s optional sampling theorem states that the properties of martingales, submartingales and supermartingales generalize to stopping times. For simple stopping times, which take only finitely many values in , the argument is a relatively basic application of elementary integrals. For simple stopping times , the stochastic interval and its indicator function are elementary predictable. For any submartingale , the properties of elementary integrals give the inequality
For a set the following
is easily seen to be a stopping time. Replacing by extends inequality (1) to the following,
As this inequality holds for all sets it implies the extension of the submartingale property to the random times. This argument applies to all simple stopping times, and is sufficient to prove the optional sampling result for discrete time submartingales. In continuous time, the additional hypothesis that the process is right-continuous is required. Then, the result follows by taking limits of simple stopping times.
Theorem 1 Let be bounded stopping times. For any cadlag martingale, submartingale or supermartingale , the random variables are integrable and the following are satisfied.
- If is a martingale then,
- If is a submartingale then,
- If is a supermartingale then,